A Systematic Methodology for Analyzing Closed-Form Heston Pricer Regarding Their Accuracy and Runtime

Calibration methods are the heart of modeling any financial process. While for the Heston model (semi) closed-form solutions exist for calibrating to simple products, their evaluation involves complex functions and infinite integrals. So far these integrals can only be solved with time-consuming numerical methods. For that reason, calibration consumes a large portion of available compute power in the daily finance business and it is worth checking for the most optimal available methods with respect to runtime and accuracy.However, over the years more and more theoretical and practical subtleties have been revealed and today a large number of approaches are available, including dierent formulations of closed-formulas and various integration algorithms like quadrature or Fourier methods. Currently there is no clear indication which pricing method should be used for a specific calibration purpose with additional speed and accuracy constraints. With this publication we are closing this gap. We derive a novel methodology to systematically find the best methods for a well-defined accuracy target among a huge set of available methods. For a practical setup we study the available popular closed-form solutions and integration algorithms from literature. In total we compare 14 pricing methods, including adaptive quadrature and Fourier methods. For a target accuracy of 10-3 we show that static Gauss-Legendre are best on CPUs for the unrestricted parameter set. Further we show that for restricted Carr-Madan formulation the methods are 3.6x faster. We also show that Fourier methods are even better when pricing at least 10 options with the same maturity but dierent strikes.